Let $α \in [0, 1/2)$ and $n, t \geq 1$ . Let $F^{(t)} (n, α)$ be the smallest $m$ such that we can $2$ -colour the edges of the complete $t$ -uniform hypergraph on $n$ vertices such that if $X \subseteq [n]$ with $∣ X ∣ \geq m$ then there are at least $α (t ∣ X ∣)$ many $t$ -subsets of $X$ of each colour. For fixed $n, t$ as we change $α$ from $0$ to $1/2$ does $F^{(t)} (n, α)$ increase continuously or are there jumps? Only one jump?

Worked, still open.

combinatorics · open · prize $500 · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (E=\binom{n}{t}) for the number of $t$-edges in (K_n^{(t)}), and (as is standard) take (m\in{1,2,\dots,n}).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

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